Pathological mathematics

[ughaibu]

I'm looking for an overview and general discussion of the history and theory of pathological mathematics, any suggestions or recommendations?

[Turtle]

I'm looking for an overview and general discussion of the history and theory of pathological mathematics, any suggestions or recommendations?

Well, I had to look up 'pathological mathematics' at Wicky; have you tried there yet? Do you have some other source already, or just throwing out the phrase to see who bites?

In looking over the description, I have to say the Strange numbers and their kin discussed here are pathological as well as strange.:) There is at least considerable discussion there as to the nature of their pathology if not the phenomenon itself.:cup: :eek:

[ughaibu]

Turtle: Thanks. Yes I've read the Wikipedia page and a few other brief articles uncovered by searches, these are thought provoking but dont explore the ramifications in any depth.

[Qfwfq]

I'd say that wiki is about the word pathological in the context of mathematics. There is not exactly such a topic as "pathological mathematics" it is just a manner of indicating cases which can't be treated in the same manner as those under consideration. It is just an expression, really.

 

One might say the function defined as 1/x is pathological in x = 0 which therefore does not belong to the function's domain. The number 1 is not considered prime, just for the convenience of the prime number theorem. Examples abound.

[ughaibu]

Qfwfq: Doesn't this imply uncertainty, that mathematics cant guarantee satisfiability? Isn't pathological mathematics important for proof theorists?

[CraigD]

It’s important to note, I think, that the adjective “pathological” when applied to math is subjective. Taking 2 of the examples from the wikipedia article,

• Irrational numbers such as [Latex]\sqrt{2}[/Latex], while troubling to contemporaries of Pythagoras (and, at some point in their education, to most serious students), are a useful addition to math, allowing the generation of a useful numeric sets such as surds and real numbers
  
• Fractal generating functions, such as the famous Mandelbrot set, were commonly deemed “pathological”, until experience with them yielded useful applications
  

In this light, “pathological” is more a term of the state of development of a branch of math and a community of mathematicians, than an intrinsic property of the math, similar to the term “adolescent” applied to a human.

[Qfwfq]

Doesn't this imply uncertainty, that mathematics cant guarantee satisfiability?

No, I wouldn't say so. It's typically a matter of specifying which properties something must have in order for a thesis to hold. The hypothesis might have to specify things such as "at least continuous and differentiable" or "holomorphic in a simply connected domain" or whatever.

[ughaibu]

Thanks for the replies.

Would you class Cardano's discovery of complex numbers as a pathological effect, or as a form of fallacy? Generally speaking, is the distinction between fallacy and pathology clear?

[Qfwfq]

First, I tend to say 'definition' rather than 'discovery'. It was an idea and it hadn't previously been thought of because not just everybody would have, though the modern approach to math is that of saying "Well, if no number from [math]\norm -\infty[/math] to [math]\norm +\infty[/math] is the root of -1, then it's a number in a wider set" instead of "then it doesn't exist".

 

I'm not sure what you mean by "as a pathological effect, or as a form of fallacy" but certainly there is no fallacy about complex numbers. The pathology is that of the real field, in which a polynomial of degree n may have less than n roots (even counting multiplicity). The complex field doesn't have this problem, it is algebraically complete, a perfect little field.

[ughaibu]

Qfwfq: What I meant was that by using complex (or perhaps imaginary?) numbers, Cardano solved the problem of finding two numbers whose sum is ten and whose product is forty, on the face of it this might appear to be a result in the same class as various "false proofs" demonstrating that one equals zero or similar, on the other hand it might be a "pathological" result consequent to the definitions failing to exclude roots of negative numbers, then again, it might be classed as something else. I'm wondering how a distinction is drawn between pathological results and false results, is it entirely dependent on the definitions?

[Qfwfq]

Well, no, it isn't in the same class as "false proofs" because there isn't a self contradiction such as "one equals zero", there is only the pathology of the real field which I mentioned. Solving

 

[math]a+b=10[/math]

[math]ab=40[/math]

 

by a simple substitution is equivalent to solving

 

[math]a^2-10a+40=0[/math]

 

which means finding a root of this equation of degree two. Even when only real numbers were considered, it simply appeared that roots did not always exist but this isn't a self contradiction. The fact that the discriminant is -60 appeared to mean that it has no roots, we now simply specify that it has no real roots.

[ughaibu]

Okay, thanks.